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Crouching AGM, hidden modularity

Crouching AGM, hidden modularity
Shaun Cooper, Jesús Guillera, Armin Straub, Wadim Zudilin — to appear as Chapter 9 of the book: Frontiers in Orthogonal Polynomials and q-Series (World Scientific) — Editors: Z. Nashed and X. Li — 2018

Abstract

Special arithmetic series \(f(x)=\sum_{n=0}^\infty c_nx^n\), whose coefficients \(c_n\) are normally given as certain binomial sums, satisfy `self-replicating' functional identities. For example, the equation $$ \frac1{(1+4z)^2}\,f\biggl(\frac z{(1+4z)^3}\biggr) =\frac1{(1+2z)^2}\,f\biggl(\frac{z^2}{(1+2z)^3}\biggr) $$ generates a modular form \(f(x)\) of weight 2 and level 7, when a related modular parametrization \(x=x(\tau)\) is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing \(\pi\) and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.

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BibTeX

@article{crouching-agm-2018,
    author = {Shaun Cooper and Jes\'us Guillera and Armin Straub and Wadim Zudilin},
    title = {Crouching {AGM}, hidden modularity},
    journal = {to appear as Chapter 9 of the book: Frontiers in Orthogonal Polynomials and q-Series (World Scientific)},
    year = {2018},
    doi = {10.1142/10677},
}